Fractional Differentiability for Solutions of Nonlinear Elliptic Equations

We study nonlinear elliptic equations in divergence form(Formula presented.) When (Formula presented.) has linear growth in Du, and assuming that (Formula presented.) enjoys (Formula presented.) smoothness, local well-posedness is found in (Formula presented.) for certain values of (Formula presented.) and (Formula presented.). In the particular case (Formula presented.), G = 0 and (Formula presented.), (Formula presented.), we obtain (Formula presented.) for each (Formula presented.). Our main tool in the proof is a more general result, that holds also if (Formula presented.) has growth s−1 in Du, 2 ≤ s ≤ n, and asserts local well-posedness in Lq for each q > s, provided that (Formula presented.) satisfies a locally uniform VMO condition